18.783 Elliptic Curves Lecture 1 - MIT Mathematics
18.783 Elliptic Curves
Lecture 1
Andrew Sutherland
February 8, 2017
What is an elliptic curve?
The equation
x2
a2
+
y2
b2
= 1 defines an ellipse.
An ellipse, like all conic sections, is a curve of genus 0.
It is not an elliptic curve. Elliptic curves have genus 1.
The area of this ellipse is ¦Ðab. What is its circumference?
The circumference of an ellipse
p
Let y = f (x) = b 1¡Ì? x2 /a2 .
Then f 0 (x) = ?rx/ a2 ? x2 , where r = b/a < 1.
Applying the arc length formula, the circumference is
Z ap
Z ap
4
1 + f 0 (x)2 dx = 4
1 + r2 x2 /(a2 ? x2 ) dx
0
0
With the substitution x = at this becomes
Z 1r
1 ? e2 t2
4a
dt,
1 ? t2
0
¡Ì
where e = 1 ? r2 is the eccentricity of the ellipse.
This is an elliptic integral. The integrand u(t) satisfies
u2 (1 ? t2 ) = 1 ? e2 t2 .
This equation defines an elliptic curve.
An elliptic curve over the real numbers
With a suitable change of variables, every elliptic curve with real
coefficients can be put in the standard form
y 2 = x3 + Ax + B,
for some constants A and B. Below is an example of such a curve.
y 2 = x3 ? 4x + 6
over R
An elliptic curve over a finite field
y 2 = x3 ? 4x + 6
over F197
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